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Baire Spaces and Hyperspace Topologies Revisited Appl. Gen. Topol. 15, no. 1 (2014), 85-92 @ doi:10.4995/agt.2014.1897 c AGT, UPV, 2014 Baire spaces and hyperspace topologies revisited Steven Bourquin and Laszl´ o´ Zsilinszky Department of Mathematics and Computer Science, The University of North Carolina at Pem- broke, Pembroke, NC 28372, USA ([email protected], [email protected]) Abstract It is the purpose of this paper to show how to use approach spaces to get a unified method of proving Baireness of various hyperspace topologies. This generalizes results spread in the literature including the general (proximal) hit-and-miss topologies, as well as various topologies gener- ated by gap and excess functionals. It is also shown that the Vietoris hyperspace can be non-Baire even if the base space is a 2nd countable Hausdorff Baire space. 2010 MSC: Primary 54B20; Secondary 54E52, 54A05 Keywords: Baire spaces, approach spaces, (proximal) hit-and-miss topolo- gies, weak hypertopologies, Banach-Mazur game 1. Introduction A topological space is a Baire space [7] provided countable collections of dense open subsets have a dense intersection. Baireness is one of the weakest completeness properties, and yet, it has fundamental applications throughout mathematics. This is why there has been interest in investigating Baireness, along with other completeness properties, in the theory of hyperspace topolo- gies which in turn have applications in various branches of mathematics on their own ([1], [12]). In the present paper we will continue in this research by exhibit- ing some common features of the plethora of studied hyperspaces as far as their Baireness is concerned. Indeed, following the unified exposition of hyperspace topologies introduced in [20], we prove Baireness of these general hyperspaces in Received 16 November 2013 – Accepted 13 February 2014 S. Bourquin and L. Zsilinszky one theorem, thereby generalizing and extending several results about specific hyperspaces spread in the literature ([1],[14],[19],[23],[21],[3],[4],[9]). In hyperspace theory, it is customary to assume that the base space is Haus- dorff (or at least T1), since then singletons are closed, and the base space embeds into the hyperspace. It was observed that imposing separation axioms on the base space is frequently not necessary to obtain results on hypertopologies (see [6], [19], [24], [10]), which is the case throughout this paper as well. At the end, an example is provided that shows the limitations of Baireness results for the Vietoris topology. 2. Preliminaries Throughout the paper ω stands for the non-negative integers, P(X) for the power set, CL(X) for the nonempty closed subsets of a topological space X, and Ec for the complement of E ⊂ X in X. The general description of the hyperspaces we will use was given in [20], here we just provide the definitions so the paper is self-contained, more detail can be found in [20], and the references therein. Suppose that (X,δ) is an approach space [13], i.e. X is a nonempty set and δ : X ×P(X) → [0, ∞] is a so-called distance (on X) having the following properties: (D1) ∀x ∈ X : δ(x, {x})=0 (D2) ∀x ∈ X : δ(x, ∅)= ∞ (D3) ∀x ∈ X ∀A, B ⊂ X : δ(x, A ∪ B) = min{δ(x, A),δ(x, B)} (D4) ∀x ∈ X ∀A ⊂ X ∀ε> 0 : δ(x, A) ≤ δ(x, Bε(A)) + ε, where Bε(A) = {x ∈ X : δ(x, A) ≤ ε}; we will also use the notation Sε(A) = {x ∈ X : δ(x, A) <ε}. Every approach space (X,δ) generates a topology τδ on X defined by the closure operator: A¯ = {x ∈ X : δ(x, A)=0}, A ⊂ X. The functional D : P(X) ×P(X) → [0, ∞] will be called a gap provided: (G1) ∀A, B ⊂ X : D(A, B) ≤ inf δ(a,B) a∈A (G2) ∀x ∈ X ∀A ⊂ X : D({x}, A) = inf δ(y, A) y∈{x} (G3) ∀A, B, C ⊂ X : D(A ∪ B, C) = min{D(A, C),D(B, C)}. In the sequel (unless otherwise stated) X will stand for an approach space (X,δ) with a gap D (denoted also as (X,δ,D)). Remark 2.1. (1) Examples of distances: • [13] Let X be a topological space. For x ∈ X and A ⊂ X define 0, if x ∈ A,¯ δt(x, A)= (∞, if x∈ / A.¯ c AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 86 Baire spaces and hyperspace topologies revisited Then δt is a distance on X, and τδt coincides with the topology of X. • Let (X, U) be a uniform space. Then U is generated by the family D of uniform pseudo-metrics on X such that d ≤ 1 for all d ∈ D, and d1, d2 ∈ D implies max{d1, d2} ∈ D. Then δu(x, A) = sup d(x, A), x ∈ X, A ⊂ X d∈D defines a (bounded) distance on X and τδu coincides with the topology induced by U on X. • Let (X, d) be a metric space. Then δm(x, A)= d(x, A) is a distance on X and τδm is the topology generated by d on X. (2) Examples of gaps: • For an arbitrary approach space (X,δ) D(A, B) = inf δ(a,B) a∈A is clearly a gap on X. This is how we are going to define the gap Dt (resp. Dm) in topological (metric) spaces using the relevant distance δt (resp. δm). • Let (X, U) be a uniform space generated by the family D of uni- form pseudo-metrics on X bounded above by 1. For A, B ⊂ X define Du(A, B) = sup inf d(a,B). d∈D a∈A Then Du is a gap on X. (3) Excess functional: for all A, B ⊂ X define the excess of A over B by e(A, B) = sup δ(a,B). a∈A The symbols et,eu,em will stand for the excess in topological, uniform and metric spaces, respectively defined via δt,δu,δm. 3. Hyperspace topologies For E ⊂ X write E− = {A ∈ CL(X): A ∩ E 6= ∅}, E+ = {A ∈ CL(X): A ⊂ E} and E++ = {A ∈ CL(X): D(A, Ec) > 0}. In what follows ∆1 ⊂ CL(X) is arbitrary and ∆2 ⊂ CL(X) is such that ∀ε> 0 ∀A ∈ ∆2 ⇒ Sε(A) is open in X. D D ∅ k+1 k+1 2k+2 Denote = (∆1, ∆2)= k∈ω(∆1 ∪{ }) × (∆2 ∪{X}) × (0, ∞) . Whenever referring to some S,T ∈ D, we will assume that for some k,l ∈ ω S S = (S0,...,Sk; S˜0,..., S˜k; ε0,...,εk;˜ε0,..., ε˜k) T = (T0,...,Tl; T˜0,..., T˜l; η0,...,ηl;˜η0,..., η˜l). c AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 87 S. Bourquin and L. Zsilinszky For S ∈ D denote c ˜ M(S)= (Bεi (Si)) ∩ Sε˜i (Si) and i k \≤ ∗ S = {A ∈ CL(X): D(A, Si) >εi and e(A, S˜i) < ε˜i}. i k \≤ For U0,...,Un ∈ τδ \{∅} and S ∈ D denote − ∗ (U0,...,Un)S = Ui ∩ S , i n \≤ [U0,...,Un]S = (Ui ∩ M(S)) × M(S). i n i>n Y≤ Y It is easy to see that the collections ∗ B = {(U0,...,Un)S : U0,...,Un ∈ τδ \{∅},S ∈ D,n ∈ ω}, B = {[U0,...,Un]S : U0,...,Un ∈ τδ \{∅},S ∈ D,n ∈ ω} form a base for topologies on CL(X) and Xω, respectively; denote them by τ ∗, and τ, respectively. Remark 3.1. Note that τ is a “pinched-cube” topology as defined in [3], [23]; indeed, we just need to take ∆ = {M(S)c : S ∈ D}. Remark 3.2. • Let (X, τ) be a topological space. Let ∆1 = ∆ and ∆2 = {X}. Then c + for B ∈ ∆ and ε,η > 0, {A ∈ CL(X): Dt(A, B) > ε} = (B ) and ∗ + {A ∈ CL(X): et(A, X) < η} = CL(X). Thus τ = τ is the general hit-and-miss topology on CL(X) (see [17], [1], [8], [19], [3]). Choosing ∆ = CL(X) we get the most studied hit-and-miss topology, the so- called Vietoris topology τV (cf. [15], [5], [1]); a typical base element for τV is (U0,...,Un)= {A ∈ CL(X): A ⊆ Ui and A ∩ Ui 6= ∅ for all i ≤ n}. i n [≤ • Let (X, U) be a uniform space. Let ∆1 = ∆ and ∆2 = {X}. Then c ++ for B ∈ ∆ and ε,η > 0, {A ∈ CL(X): Du(A, B) > ε} = (B ) ∗ ++ and {A ∈ CL(X): eu(A, X) < η} = CL(X). Thus τ = τ is the proximal hit-and-miss topology on CL(X) (see [1], [19], [3]). • Let (X, d) be a metric space. Let ∆1, ∆2 ⊂ CL(X) be such that ∆1 contains the singletons. Then τ ∗ coincides with the weak hypertopology τweak generated by gap and excess functionals (see [2], [9], [20]). As we indicated in the Introduction, we do not assume any separation axiom on X, however, the following property seems to be necessary: we will say that (X,δ,D) has property (P) provided ∀x ∈ X ∀k ∈ ω ∀A0,...,Ak ∈ CL(X) ∀ε0,...,εk > 0 ∃y ∈ {x} : δ(x, Ai) >εi =⇒ D({y}, Ai) >εi for all i ≤ k. c AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 88 Baire spaces and hyperspace topologies revisited This property is satisfied in uniform and metric spaces, and a topological space X has property (P) iff X is weakly-R0 [24], i.e. for all open U ⊆ X and x ∈ U there is a y ∈ {x} with {y} ⊂ U iff each nonempty difference of open sets contains a nonempty closed set. We will say that the family D is weakly quasi-Urysohn [20] provided for all ∗ ∅ 6= (U0,...,Un)S ∈ B there is a T ∈ D such that ∅ 6= (U0,...,Un)T ⊂ (U0,...,Un)S and (*) ∀E countable : E ⊂ M(T )=⇒ E ∈ S∗.
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